maths

 

 

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MAB533 - Statistical Techniques

Assignment 2

Survival Analysis

Due date: 5pm, 7th May, 2014

When preparing your final submission, for each question, you are required to include all relevant R-code. Marks will be deducted if any code you supply is not clear, concise, efficient, error free and well commented.

A study was undertaken to investigate routine bathing and body cleansing as treatments for wounds of burns victims. Interest was in determining whether there is a difference between these two treatments in terms of the time to excision (a surgical procedure that doctors recommend for deep burns that cannot heal on their own). A description of the data collected is given below.

• ‘treat’: Treatment 0=routine bathing, 1=body cleansing;

• ‘gender’: Gender 0=male 1=female;

• ‘race’: Race 0=nonwhite 1=white;

• ‘perc’: Percentage of total surface area burned;

• ‘head’: Burn site indicator: head 1=yes, 0=no;

• ‘buttock’: Burn site indicator: buttock 1=yes, 0=no;

• ‘trunk’: Burn site indicator: trunk 1=yes, 0=no;

• ‘upper’: Burn site indicator: upper leg 1=yes, 0=no;

• ‘lower’: Burn site indicator: lower leg 1=yes, 0=no;

• ‘type’: Type of burn 1=chemical, 2=scald, 3=electric, 4=flame;

• ‘y’: Time to excision or on study time (months) ; and

• ‘cen’: Excision indicator 1=yes 0=no .2

1. Initially, suppose the data are split into the two treatment groups based on the value of ‘treat’ above, and (for the moment) ignore all other covariate/factor information. Assume proportional hazards between the two groups and that the survival times in treatment group 0 are distributed as Weibull W(λ,γ).

(a) Define the hazard function for individuals in treatment group 0.

(b) What is the distribution of survival times for individuals in treatment group 1, and what is the hazard function for these individuals?

(c) Suppose the survival times for individuals in treatment group 0 are dis- tributed W(λ, 1). By hand, find the MLEs (and standard errors) for λ and the proportionality parameter ψ (show all working).

(8 marks)

2. Now consider all available covariate/factor information when answering the fol- lowing questions.

(a) Using the four step model selection procedure based on the −2 log L̂ statis- tic (as discussed in lectures), determine which covariates/factors should be included into the hazard function under a fully parametric Weibull propor- tional hazards model. Assume a significance level of α = 0.10. Please ensure you provide step-by-step details of how your final model was derived.

(b) Given your final model, is there a difference between the two treatment types in terms of time to excision? If so, which treatment prolongs time to exci- sion?

(c) Given your final model, define the ith individuals hazard function, remem- bering to define which group of individuals the baseline hazard refers to.

(d) Estimate the median survival time for an individual with treat = 0, gender = 0, race = 0, perc = 15, head = 0, buttock = 0, trunk = 1, upper = 1, lower = 0 and type = 2, and interpret.

   

 

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1. Initially, suppose the data are split into the two treatment groups based on the value of ‘treat’ above, and (for the moment) ignore all other covariate/factor information. Assume proportional hazards between the two groups and that the survival times in treatment group 0 are distributed as Weibull W(λ,γ).

(a) Define the hazard function for individuals in treatment group 0.

(b) What is the distribution of survival times for individuals in treatment group 1, and what is the hazard function for these individuals?

(c) Suppose the survival times for individuals in treatment group 0 are dis- tributed W(λ, 1). By hand, find the MLEs (and standard errors) for λ and the proportionality parameter ψ (show all working).

(8 marks)

2. Now consider all available covariate/factor information when answering the fol- lowing questions.

(a) Using the four step model selection procedure based on the −2 log L̂ statis- tic (as discussed in lectures), determine which covariates/factors should be included into the hazard function under a fully parametric Weibull propor- tional hazards model. Assume a significance level of α = 0.10. Please ensure you provide step-by-step details of how your final model was derived.

(b) Given your final model, is there a difference between the two treatment types in terms of time to excision? If so, which treatment prolongs time to exci- sion?

(c) Given your final model, define the ith individuals hazard function, remem- bering to define which group of individuals the baseline hazard refers to.

(d) Estimate the median survival time for an individual with treat = 0, gender = 0, race = 0, perc = 15, head = 0, buttock = 0, trunk = 1, upper = 1, lower = 0 and type = 2, and interpret.

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(14 marks)

Suppose you are to plan a follow up study, and there is interest in determining the chance of declaring the observed difference between the two treatments as significant at the 5% significance level with a minimum treatment effect of θR = 0.12 (lower end of the 95% confidence interval for this parameter) when only 300 patients can be enrolled. Unfortunately we do not have a formula to calculate the specified probability. You will therefore need to use simulation techniques to approximate an answer.

In answering this question via simulation, assume the following:

• The new study will run for 24 months only;

• Any patient who has not experienced an event when the study finishes will yield right censored data;

• All patients have been recruited at time zero months such that there is no recruit period (every patient is ready to go at time zero);

• No patient will drop out of the study unless an event has been experienced;

• The treatments will be allocated at random with equal probability; and

• A reasonable distribution for covariates/factors that appear in your final model. For example, if gender is an important variable in your model then you could assume that (on average) 77.92% of patients in your new study will be male as this is the proportion of males in the current data set.

List any other assumptions you have made. Interpret the probability that you have calculated and the implications for this future planned study.

(12 marks)

Additional question for MAN students.

Search the literature for an example of where one or more of the survival analysis techniques covered in MAN533 have been used. Determine the aim of the research, the statistical approach used for inference and the relevant statistical conclusions. A paragraph or two should suffice. Ensure a full reference is provided. (5 marks)